zbMATH — the first resource for mathematics

Global solvability for a special class of vector fields on the torus. (English) Zbl 1108.35026
Berhanu, Shiferaw (ed.) et al., Recent progress on some problems in several complex variables and partial differential equations. Papers presented at the international conferences on partial differential equations and several complex variables, Wuhan, China, June 9–13, 2004 and on complex geometry and related fields, Shanghai, China, June 21–24, 2004. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3921-7/pbk). Contemporary Mathematics 400, 11-20 (2006).
Summary: We study the global solvability of a class of complex vector fields on the two-torus. For \(L=\partial/\partial t+(a(x)+ib(x))\partial/\partial x\), \(a,b\in C^\infty(\mathbb{T}^1;\mathbb{R})\), we show that a necessary condition for \(L\) to be strongly solvable is that each zero of \(a+ib\) is of finite order. We say that \(L\) is strongly solvable if the image of operator \(L:C^\infty (\mathbb{T}^2)\to C^\infty(\mathbb{T}^2)\) is closed and has finite codimension. One of the main points of our work is to shed light on the interplay between the orders of vanishing of \(a\) and \(b\) at each common zero, which is crucial for strong solvability of \(L\).
For the entire collection see [Zbl 1086.32001].

35F05 Linear first-order PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B10 Periodic solutions to PDEs
58J99 Partial differential equations on manifolds; differential operators